Let \(Q\)  stands for the total number of qubits in the utilized quantum circuit and \(\vert \delta _{q} \rangle \) , \(q=1,...,Q\) , stands for the quantum state of the \(q^{th}\)  qubit with two possible values, \(\delta _q =0\)  or \(\delta _q = 1\)  in Dirac notations [1]}. \(\vert 0 \rangle ^{\otimes Q}\)  refers to the tensor product of \(\vert 0 \rangle \)  of \(Q\)  times, which is equivalent to \(\vert 0 \rangle \otimes \vert 0 \rangle \otimes \cdot \cdot \cdot \otimes \vert 0 \rangle \) , as the initial quantum state. More specifically, we consider a quantum encoding by using a parametric quantum circuit, with a unitary operator \(U(\textbf {x})\)  of \(Q\)  qubits, such that the input vector \(\textbf {x}\)  is encoded into an embedded quantum state for the entire circuit, \(\vert \phi (\textbf {x}) \rangle \) . In doing so, \(\vert \phi _q(\textbf {x}) \rangle \)  represents the quantum state of \(q^{th}\)  qubit after encoding.
Next, we take a quantum measurement of \(\vert \phi (\textbf {x})\rangle \)  based on the Pauli-Z axis for many times [1]}, and then averages the measured values to produce a vector output, \(\textbf {y}\) , with the same dimension of the original input vector \(\textbf {x}\) . Now we can project the associated embedded quantum states \(\vert \phi (\textbf {x}) \rangle \)  back to the original classical space, such that for each pair (\(\textbf {y}_{i}, \textbf {y}_{j}\) ), a kernel can be redefined as:
\(K_{C}(\textbf {y}_{i}, \textbf {y}_{j}) \rightarrow ~&K_{Q}(\textbf {x}_{i}, \textbf {x}_{j}) = \left|\langle \phi (\textbf {x}_{i}) \vert \phi (\textbf {x}_{j}) \rangle \right|,\) 
